Question

Derive expression for the terminal velocity of a small body falling through a viscous liquid.

Answer 1

When a body falls in a viscous medium, it carries with its layers of fluid which are in body's contact where as the layers of fluid in contact with the stationary surface remain almost at rest. The layers of fluid destroys the relative motion and motion of the body is thus opposed. The viscous drag increases with velocity of the body till viscous drag and upthrust of the body are together equal to the weight of the body which acts downwards.

When there is no net force, the body moves with the uniform velocity. This velocity is called terminal velocity.

Stoke showed that the retarding force F due to viscous drag for a spherical body of radius r that moves with a velocity v in a fluid, with coefficient of viscosity $\eta$, is given as :

$F=6\pi \eta rv$

This expression is known as Stoke's Law.

Derivation with help dimensions :

$F\alpha v$, velocity

$F\alpha r$, radius of the body;

$F\alpha \eta$, coefficient of viscosity of the fluid.

$\Rightarrow F=A{\eta }^a{r}^b{v}^c$

where A is a constant with no dimensions. Putting the dimensions,

$\left[ML{T}^{-2}\right]={\left[M{L}^{-1}{T}^{-2}\right]}^a{\left[L\right]}^b{\left[L{T}^{-1}\right]}^c$

or $\left[ML{T}^{-2}\right]=\left[M^a{L}^{-a+b+c}{T}^{-a-c}\right]$

Comparing the two sides, we get

a = 1, b = 1 and c = 1

$\therefore F=A\eta rv$

A was determined to be $6\pi$.

$\therefore F=6\pi \eta rv$

Derivation for terminal velocity :

Let $p$ be the density of the body and $\rho$ be the density of the medium. Then,

Weight of body = $\frac{4}{3}\pi r^{3}pg$ ........(11.4)

Upthrust of body = $\frac{4}{3}\pi r^3\sigma g$ ........(11.5)

Resultant force (downwards) = $\frac{4}{3}\pi r^3(p-\sigma )g$

In equilibrium,

$6\pi \eta rv=\frac{4}{3}\pi r^3(p-\sigma )g$ ......(11.6)

$\Rightarrow v=\frac{2}{9}{r}^2\frac{(p-\sigma )}{\eta }g$ .....(11.7)

This is the formula for terminal velocity.

It is apparent from the above expression that the terminal velocity is (a) directly proportional to the square of the radius of body, (b) directly proportional to the densities of the body and the medium, (conversely proportional to the coefficient of viscosity of the medium.

Variation of viscosity with temperature and pressure effect of temperature on viscosity :

(i) When a liquid is heated then the kinetic energy of it's molecules increases and the intermolecular force between them is decreases. Hence the viscosity of a liquid decreases with the increase in it's temperature.

(ii) Viscosity of gases is due to the diffusion of molecules from one moving layer to another. But the rate of diffusion of a gas is directly proportional to the square root of the temperature. So the viscosity of a gas increase with it's temperature.

Effect of pressure :

(i) Except the water of the viscosity of liquid increases with the increase in pressure. In case of water, the viscosity decrease with increase in pressure.

(ii) The viscosity of gas is independent of pressure.